Multiply :
`-(3)/(2)x^(5)y^(3)` and `(4)/(9)a^(2)x^(3)y`

Answer

To multiply the expressions \(-\frac{3}{2}x^{5}y^{3}\) and \(\frac{4}{9}a^{2}x^{3}y\), we will follow these steps:

### Step 1: Write the expressions together
We start by writing the two expressions that we need to multiply:
\[ -\frac{3}{2}x^{5}y^{3} \times \frac{4}{9}a^{2}x^{3}y \]

### Step 2: Multiply the coefficients
Next, we multiply the numerical coefficients:
\[ -\frac{3}{2} \times \frac{4}{9} \]
Calculating this gives:
\[ -\frac{3 \times 4}{2 \times 9} = -\frac{12}{18} = -\frac{2}{3} \]

### Step 3: Multiply the variables
Now we will multiply the variable parts. We will combine the \(x\) terms and the \(y\) terms:
- For \(x\):
\[ x^{5} \times x^{3} = x^{5+3} = x^{8} \]
- For \(y\):
\[ y^{3} \times y = y^{3+1} = y^{4} \]

### Step 4: Combine all parts
Now we can combine all the parts together:
\[ -\frac{2}{3}a^{2}x^{8}y^{4} \]

### Final Answer
Thus, the final result of the multiplication is:
\[ -\frac{2}{3}a^{2}x^{8}y^{4} \]

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